h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I don't know about how long-lived the band Chumbawumba was in England, but over here in the US, they had one fantastically huge hit, "Tubthumping." You know the one:

I get knocked down,
but I get up again,
they're never gonna keep me down

It was all over the radio, and suddenly, they evaporated from the public eye, like most bands that insist on repeating the same phrase 100's of times, and calling it a song. (Of course, the famous exception to this rule is The Beatles song "Carry That Weight")

This is a great example of the first and second sections of this entry - Chumbawamba had a couple of chart hits in the UK but are mainly an underground album / live band. So they are known for just one song, but had one other that people forget ('Amnesia' "Do you sufer from long-term memory loss? / I can't remember"), and they also had just one hit in one country, but more than one in their own country.